Yao Shunyu's team at Tsinghua University proposes the Algebraic World Model (AWM), using Lean 4 to formally prove the rules of games

The research team led by Professors Andrew Yao and Yang Yuan at Tsinghua University has produced a major project called the "Algebraic World Model" (AWM).

People have long liked to ask one question:

Embarrassingly, training data is easily contaminated, making it hard to recreate a scenario like this: build a completely unfamiliar world and let AI observe and experiment from scratch.

There is a game set called ARC-AGI-3, which is now the place where major frontier models show off their strength. The defining feature of this game is that it tells you nothing—no rules, no goals.

It simply lets you figure things out on your own.

Top players like GPT Astra, paired with the Provider Adapter framework, can score 99.9% here.

While playing the game, it organizes the game mechanics into a pile of symbolic rules.

Dense and packed, the information density is very high, and it is far more precise than other models.

But in essence, this thing is like Tycho's experimental notebook.

Tycho, Kepler's teacher, was an observation fanatic who recorded enormous amounts of planetary motion data with frightening precision.

But he also only recorded phenomena.

The one who truly discovered the three laws of planetary motion from these data and revealed the essential laws behind them was Kepler.

The rules generated by Astra are Tycho's notebook, not Kepler's laws. They only describe phenomena and do not grasp the essence of how this mini game world operates.

There will still be "hallucinations."

What problems arise from recording only phenomena?

The researchers took one of the games, called sc25, and modified it slightly, setting the initial step count to a number that makes clearing it fundamentally impossible.

For example, in the fourth level, you originally need at least 19 steps to win, but they gave you only 18.

Then they had Astra play it.

As a result, Astra tried 8 rounds, became a bit frustrated, and then asserted: 18 steps definitely won't work; I estimate it takes at least 22 steps to win.

Yes, not only did it fail to find the correct answer, it also gave an absurdly wrong figure of 22 steps.

This was not an isolated case. In 12 similar tests, the top GPT-6 Astra and Claude Opus 5.5 combined judged "impossible" only 5 times, and 4 of those times the minimum step count they gave was still wrong.

In a game, this kind of hallucination at most trips you up.

At this point, the power of algebra is needed. That is, the Algebraic World Model (AWM) developed by the Tsinghua team mentioned earlier.

It abstracts all objects, relations, and rules in the game into an "algebraic world." Then, using the large model's now super-strong mathematical ability, it understands this world.

The process is roughly three steps:

Exploration: first play a few times and write the observed phenomena as executable Python code, as preliminary "hypotheses" or "laws."

Distillation: use new observations to verify and revise these hypotheses, then refine them into more general objects and rules.

Proof: finally, formalize these laws into a language called Lean 4, turning them into logically rigorous mathematical assertions that machines can automatically check.

Put plainly, it moves from exploratory trial and error to modeling and formal proof.

How well does it work?

Faced with the unsolvable level just mentioned, AWM based on the Opus 5.5 model needed only two attempts to directly prove that this level is unsolvable under the current step count, and that only one more step would be needed to win.

Besides being able to judge "impossible," AWM is also stronger at clearing levels. Across the entire ARC-AGI-3 test set, using Opus 5, it cleared all 183 levels with a score of 99.9%, while the average number of actions per clear was only 43% of the human baseline.

It is the most efficient among all currently known systems.

So how exactly is AWM implemented?

You may have heard of algebraic geometry and algebraic topology. Simply put, algebraic geometry connects algebraic equations with geometric shapes, while algebraic topology uses algebraic tools to study properties of spaces such as connectivity and holes.

By algebraizing shapes and spaces, many questions that are hard to answer by observation alone can be solved. Mathematicians created algebraic geometry and algebraic topology because geometry and topology are the worlds they care about.

Now, with the power of AI, AWM can extend this path to more unfamiliar games, interactive environments, and even real-world tasks: discover laws through exploration, distill structures from laws, and then use mathematics to understand these structures.

Think like Newton

The idea behind this actually runs through the entire history of science.

From universal gravitation and Maxwell's equations to relativity, humanity's top intelligence has always been doing one thing: extracting the most basic laws from complex phenomena and describing the world in the unified language of mathematics.

What AWM wants to do is teach AI this same method.

To achieve this goal, it drew inspiration from a branch of mathematics called "category theory." This thing sounds terrifying and is known as "the mathematics of mathematics," because what it studies is not some specific field of mathematics, but the common structures behind different mathematical theories.

AWM mainly uses two tools from it, called "projective limits" and "inductive limits," which are essentially two complementary ways of thinking:

Defining (projective limit): piecing together a bunch of existing concepts and their relations to jointly define something new. You can understand it as decomposition and combination of concepts.

Finding commonality (inductive limit): observing a bunch of different things, finding what they have in common, and then unifying them into a more abstract framework. You can understand it as generalization and specialization of concepts.

By going back and forth between these two approaches, AWM can continuously iterate the initially scattered and chaotic Python hypotheses, making them increasingly concise, clear, and consistent, ultimately forming a world model that can be reasoned about and proved.

Think of Newton back then: what he faced was not a problem with the answer already written out, but countless scattered observational data and phenomena. The truly great work was to find a few core concepts such as "force" and "mass" from these phenomena, and then organize them into a theoretical system that can be deduced and tested.

What AWM wants to teach AI is exactly this way of working.

Of course, it also has boundaries. Its proofs hold based on the "world it has observed." If exploration is not comprehensive enough and some key rules are missed, then the model depicts only a "local truth," not the complete world.

One is Pan Guancheng of Zhejiang University, currently on leave doing research at Tsinghua University.

The other is Tan Qixin from Tsinghua's Yao Class.

The advising team is led by senior scholars in the field:

Yang Yuan, associate professor at Tsinghua University's Institute for Interdisciplinary Information Sciences and School of Artificial Intelligence, and concurrently director of the Center for Embodied Intelligence and Intelligent Manufacturing at Shanghai Qi Zhi Institute.

Andrew Yao, winner of the 2000 Turing Award and academician of the Chinese Academy of Sciences, currently dean of Tsinghua University's Institute for Interdisciplinary Information Sciences and dean of the School of Artificial Intelligence.

In addition, he is a 2021 Kyoto Prize laureate, a foreign member of the U.S. National Academy of Sciences, and a foreign member of the American Academy of Arts and Sciences.